Symmetry 2012, 4, 26-38; doi:10.3390/sym4010026 OPEN ACCESS ISSN 2073-8994 www.mdpi.com/journal/symmetry Article of Spatial Graphs and Rational Twists along Spheres and Tori

Toru Ikeda

Department of Mathematics, Faculty of Science, Kochi University, 2-5-1 Akebono-cho, Kochi-Shi, Kochi 780-8520, Japan; E-Mail: [email protected]

Received: 14 November 2011; in revised form: 12 January 2012 / Accepted: 13 January 2012 / Published: 20 January 2012

Abstract: A symmetry of a spatial graph Γ in S3 is a finite group consisting of orientation-preserving self- of S3 which leave Γ setwise . In this paper, we show that in many cases symmetry groups of Γ which agree on a regular neighborhood of Γ are equivalent conjugate by rational twists along incompressible spheres and tori in the exterior of Γ .

Keywords: 3-; geometric topology; symmetry; finite ; spatial graph; rational twist

1. Introduction

There are several approaches to the theory of graphs embedded in the 3-sphere, which are often motivated by molecular chemistry, since the chemical properties of a molecule depend on the symmetries of its molecular bond graph (see, for example, [1]). The symmetries of an abstract graph Γ are described by . If Γ is embedded in S3, some of these automorphisms are induced from self-diffeomorphisms of S3. For example, [2–6] studied the extendabilities of the automorphisms of Γ , mainly in the case of Mobius¨ ladders, complete graphs, and 3-connected graphs. Even if the automorphisms of Γ extend to self-diffeomorphisms of S3, we face the problem of the uniqueness of the extensions. In this situation, it is enough to consider Γ to be a topological space, since we need to study self-diffeomorphisms of S3 which agree on Γ . In the case of a non-torus knot in S3, there are only finitely many conjugacy classes of symmetries (see [7,8]). For a cyclic period or a free period of a knot in S3, it is shown in [9,10] that the generated by the periodic self- of S3 defining the symmetry is unique up to conjugate in some cases. Moreover, Symmetry 2012, 4 27 the author [11] generalized this result to the case of links in S3. In this paper, we generalize these results to the case of symmetries of spatial graphs in S3. Suppose that any component of Γ is a non-trivial graph with no leaf. We see Γ as a geometric simplicial complex, and denote by |Γ | the underlying topological space of Γ . A tame embedding of |Γ | into S3 is called a spatial embedding of Γ into S3, or simply a spatial graph Γ in S3. We say that Γ is splittable if there exists a sphere in S3 disjoint from Γ that separates the components of Γ . We say that Γ is non-splittable if it is not splittable. Suppose that an incompressible torus in S3 − Γ bounds a solid torus V in S3 containing Γ . The core of V is called a companion knot of Γ if it is not ambient isotopic to Γ in V . If there is no companion knot of Γ , every incompressible torus in S3 − Γ separates the components of Γ . Let M be a 3-manifold, and X a submanifold of M. Denote by N(X) a regular neighborhood of X, and by E(X) = M − intN(X) the exterior of X. We refer to a finite G of the diffeomorphism group Diff(M) as a finite group action on M. actions G1 and on M are equivalent

(relative to X) if some h ∈ Diff(M) conjugates G1 to G2 (and restricts to the identity map on X). A G of a spatial graph Γ in S3 is a finite group action on the pair (S3,Γ ) which preserves the orientation of S3. 2 3 1 3 3 Let S be the sphere in R , and S the unit in the xy-plane in R . Denote by Rotθ ∈ Diff(R ) 2 1 1 the rotation about the z-axis through angle θ. Suppose that σn ∈ Diff(S ×I) and τn ∈ Diff(S ×S ×I), where n ∈ R, is given by σn(x, t) = (Rot2πnt(x), t) and τn(x, y, t) = (Rot2πnt(x), y, t). Let F be a 2-sided sphere or torus embedded in a 3-manifold M. Split M open along F into a (possibly disconnected) 3-manifold MF . Denote by F− and F+ the boundary components of MF originated from

F . An n-twist along F is a discontinuous map on M induced from a diffeomorphism on MF −F− which restricts to the identity map on E(F+) and the map on N(F+) conjugate to σn or τn according as F is a sphere or not. We say that the n-twist is rational if n ∈ Q. Figure 1 illustrates a of S3 with a setwise invariant sphere S, and its conjugate by a 1/2-twist along S.

Figure 1. Conjugation by a 1/2-twist along a sphere S.

Our main theorem is the following:

3 Theorem 1.1. Let Γ be a spatial graph in S with no companion knot. Suppose that G1 and G2 are symmetry groups of Γ such that

(1) G1(γ) = G2(γ) = γ for at least one component γ of Γ ,

(2) either Γ is non-splittable, or G1 and G2 are cyclic groups acting on Γ freely, and

(3) G1 and G2 agree on N(Γ ).

Then there is a finite sequence of rational twists along incompressible spheres and tori in E(Γ ) whose composition conjugates G2 to a symmetry group of Γ equivalent to G1 relative to N(Γ ). Symmetry 2012, 4 28

This paper is arranged as follows. In Section 2, we study symmetry groups of non-splittable spatial graph in terms of the equivariant JSJ decomposition of the exteriors. In Section 3, we establish a canonical version of the equivariant sphere theorem for the exteriors of spatial graphs with cyclic symmetry groups, and prove Theorem 1.1.

2. Non-splittable Case

For a non-splittable spatial graph Γ in S3 with a non-trivial symmetry group, there is a canonical method for splitting E(Γ ) equivariantly into geometric pieces by the loop theorem, the Dehn’s lemma, and the JSJ decomposition theorem (see [12–14]). Let M be a Haken 3-manifold with incompressible boundary. The JSJ decomposition theorem and Thurston’s uniformization theorem [15] assert that there is a canonical way of splitting the pair (M, ∂M) along a disjoint, non-parallel, essential annuli and tori into pieces (Mi,Fi) each of which is one of the following four types:

(1) Mi is an I-bundle over a compact surface and Fi is the ∂I-subbundle,

(2) Mi admits a Seifert fibration in which Fi is fibered,

(3) intMi admits a complete hyperbolic structure of finite volume, and

(4) the double of (Mi,Fi − intΦi) along a non-empty compact submanifold Φi of Fi is of type (3).

For a finite group action G on M, the fixed point Fix(G) of G is the set of points in M each of which has the stabilizer G. The singular set Sing(G) of G is the set of points in M each of which has a non-trivial stabilizer.

3 Lemma 2.1. Let T be a torus embedded in S . Suppose that G1 and G2 are orientation-preserving finite group actions on S3 such that

(1) G1(N(T )) = G2(N(T )) = N(T ),

(2) G1 and G2 do not interchange the components of ∂N(T ), and

(3) G1 and G2 agree on ∂N(T ).

b 3 Then a rational twist along a component of ∂N(T ) conjugates G2 to a finite group action G2 on S such b that the actions of G1 and G2 on N(T ) are equivalent relative to ∂N(T ).

Proof. It is enough by Lemma 2.4 of [11] to consider the case where the actions of G1 and G2 on N(T ) ∼ are not free. For each Gi, Theorem 2.1 of [16] implies that N(T ) = T × I admits a Gi-invariant product structure , in which Sing(Gi) ∩ N(T ) consists of I-fibers. Since each element of Gi takes a meridian of T to a meridian of T , the setwise stabilizer of each I-fiber is a trivial group or a 2-fold cyclic group.

Therefore, the quotient space N(T )/Gi admits the induced I-bundle structure over a 2-orbifold B with underlying surface F and n cone points of index two. Since T is a torus, the orbifold Euler

χorb(B) of B is calculated as follows (see [17]):

χorb(B) = χ(F ) − n/2 = 0. Symmetry 2012, 4 29

Since n > 0, F is a sphere and n = 4 holds.

Denote by pi : N(T ) → N(T )/Gi the projection map onto the quotient space for each i, and by Tt the T -fiber T × {t} in P1. Connect the four cone points on p1(T0) cyclically by a collection of arcs a¯1, a¯2, a¯3, and a¯4 with disjoint interiors. Each a¯i lifts to an essential loop ai on T0 such that ai and aj with i ≠ j are disjoint if |j − i| = 2, and otherwise ai meets aj transversally in a point. Suppose that each ai is isotopic to a loop bi on T1 along an annulus Bi saturated by I-fibers in P1, and to a loop ci on P T1 along an annulus Ci saturated by I-fibers in 2. Then the endpoints of each∪p2(ci) is connected by p (b ) with |i − j| = 0 or 2. Since the underlying surface of p (T ) is a sphere, 4 p (c ) is isotopic to ∪2 j ∪ 2 1 i=1 2 i 4 p (b ) relative to the cone points. Therefore, G ( 4 C ) is moved by an G -equivariant isotopy i=1 2 i ∪ 2 i=1 i 2 4 relative to T0 so as to agree with G1( i=1 Bi) on T1. P P → The I-bundle structures in 2 and 1 respectively induce orbifold ∪ φ1 : p2(T1) p (T ) and φ : p (T ) → p (T ) such that h¯ = φ ◦ φ setwise preserves the loop 4 p (b ). The 2 0 2 2 ∪0 2 1 2 1 i=1 2 i restriction of h¯ on 4 p (b ) is isotopic relative to the cone points to the identity map or an involution. ∪ i=1 2 i 4 P Since i=1 p2(bi) splits p2(T1) into two disks with no cone point, 2 is deformed by a G2-equivariant isotopy so that afterwards h¯ is the identity map or an involution. ¯ 1 Take an h-invariant S -bundle structure S1 on p2(T1) − p2(b1 ∪ b3) with respect to which p2(b2) and ¯ 1 p2(b4) are cross sectional, and an h-invariant S -bundle structure S2 on p2(T1)−p2(b2 ∪b4) with respect to which every fiber in S1 splits into two cross sections. Then S1 and S2 induce a G2-invariant product 1 1 ¯ structure S × S on T1. Let h: T1 → T1 be the lift of h which takes each ci to bi. Then we have h = Rot2πm × Rot2πn for some rational numbers m and n. Assume (m, n) ≠ (0, 0). Take a γ so that γm and γn are coprime . Then αγm + βγn = 1 holds for some integers α and β. Let ρ: R2 → S1 × S1 be the covering map given R2 (by ρ(x, y) = (Rot) 2πx(1, 0), Rot2πy(1, 0)). Denote by φ the linear transformation on represented by

α β −1 1 1 . Then the map ρ ◦ φ ◦ ρ ∈ Diff(S × S ) conjugates h to Rot × id 1 . Thus, h −γm γn 2π/γ S extends to 1/γ-twist τ along T1. Since h conjugates the action of G2 on T1 to itself, τ conjugates G2 to a finite subgroup of Diff(S3). Therefore, it is enough to consider the case (m, n) = (0, 0). × It is obvious that h = Rot2πk Rot2πl holds for any integers k and l. By∪ verifying that, for some∪ choice × 4 4 of k and l, the above argument applied to Rot2πk Rot2πl makes G2( i=1 Ci) isotopic to G1( i=1 Bi) relative to ∂N(T ), we may assume that they agree. P P ∩ By considering an isotopy of N(T ) relative to ∂N(T ) which takes 2 to 1 on Sing(G1) N∪(T ), we may assume that G and G agree on Sing(G ) ∩ N(T ). Note that Sing(G ) ∩ N(T ) splits G ( 4 B ) 1 ∪ 2 1 1 1 i=1 i into disks, and that G ( 4 B ) splits N(T ) into balls. Then the identity map on p (Sing(G ) ∩ N(T )) 1 i=1 i ∪ ∪ 2 2 4 → 4 extends to an orbifold ψ : p2( i=1 Ci) p1( i=1 Bi). Since the quotient space of any finite group action on D3 is isomorphic to one of the orbifolds listed on page 191 of [15], ψ and the identity map on p2(∂N(T )) extend to an orbifold isomorphism p2(N(T )) → p1(N(T )). Thus, G1 and

G2 are equivalent relative to ∂N(T ). Hence, the conclusion follows.

Lemma 2.2. Let M be a Seifert manifold in S3 with non-empty boundary, and F a non-empty closed 3 submanifold of ∂M. Suppose that G1 and G2 are finite group actions on S such that

(1) G1(M) = G2(M) = M and G1(F ) = G2(F ) = F , Symmetry 2012, 4 30

(2) G1(T ) = G2(T ) = T for at least one component T of F ,

(3) G1 and G2 induce the same on the set of the components of ∂M, and

(4) G1 and G2 agree on F .

Then there is a finite sequence of rational twists along incompressible tori in M whose composition b 3 b conjugates G2 to a finite group action G2 on S such that the actions of G1 and G2 on M are equivalent relative to F .

Proof. The case M = D2 × S1 and F = ∂M, the case M = S1 × S1 × I and F = ∂M, and the case M = S1 × S1 × I and F ≠ ∂M respectively follow from Lemma 2.1 of [11], Lemma 2.1 of this paper, and Theorem 8.1∪ of [16]. We therefore exclude these cases. Denote by k ξk the system of the exceptional fibers ξk in M. Let N(ξk) be a fibered regular neighborhood of each ξk. It follows from Theorem 2.2 of [16] that each Gi preserves some Seifert S fibration i∪of M. Then the uniqueness of a Seifert fibration of M (see VI.18.Theorem of [12]) implies that k N(ξk) is isotopic to a setwise Gi-invariant fibered regular neighborhood of the system of exceptional fibers in Si. Since Lemma 3.1 of [11] implies that the orders of the exceptional fibers are pairwise coprime, we may assume that G1(N(ξk)) = G2(N(ξk)) = N(ξk) for each k. Therefore, it is enough by Lemma 2.1 of [11] to consider the case where M is a product S1-bundle.

It follows from Theorem 2.1 of [16] that M admits a G1-invariant product structure P1. If F = ∂M,

M admits a G2-invariant product structure P2 which agrees with P1 on F (see Theorem 2.3 of [16]).

If F ≠ ∂M, we see M as the quotient of the double M of M along ∂M − F by Z2 generated by an orientation-reversing involution, and apply the same argument to the finite group action on M, which is the extension of Z2 by G2. Then we obtain a G2-invariant product structure P2 of M which agrees with

P1 on F . By the uniqueness of the S1-bundle structure of M (see VI.18.Theorem of [12]), there is a map 1 φ ∈ Diff(M) isotopic to the identity which takes the S -bundle structure induced by P1 to the 1 S -bundle structure induced by P2. Modify φ in P2 by a fiber preserving isotopy in a fibered regular neighborhood of F so as to restrict to the identity map on F . By conjugating G2 by φ, we 1 may therefore assume that P1 and P2 induce the same S -bundle structure of M.

Let p: M → B be the projection map onto the base surface B. Each Gi induces a finite group 2 2 action Gi on B. We consider B to be lying on S . Then each Gi extends to an action on S . Since

G1 and G2 agree on p(F ), the quotient spaces B/G1 and B/G2 are orbifold isomorphic to suborbifolds of the same spherical orbifold listed on page 188 of [15]. We may assume that G1 and G2 are not orientation-preserving, otherwise the conclusion follows from Lemma 3.2 and Remark 3.3 of [11]. Then 2 the assumption G1(T ) = G2(T ) = T implies that each Gi is generated by the reflection of S in a loop. ′ ′ Since G1 and G2 permute the components of ∂M similarly, ∂B consists of loops ℓ1, . . . , ℓ2k, ℓ1, . . . , ℓn such that

(1) G1 and G2 interchange ℓ2i−1 and ℓ2i for 1 ≤ i ≤ k, and ′ ≤ ≤ (2) G1 and G2 setwise preserve ℓi for 1 i n. Symmetry 2012, 4 31

′ Without loss of generality, ℓ1 = p(T ). Denote by Fix(Gi) the fixed point circle of the action of each 2 Gi on S . Suppose that each Fix(Gi) is equipped with an orientation, and splits B into two pieces Bi,1 ′ ∩ ′ ∩ ′ ∩ ′ ∩ and Bi,2 so that ℓ1 B1,1 = ℓ1 B2,1 and ℓ1 B1,2 = ℓ1 B2,2. We may assume without loss of ⊂ ⊂ ≤ ≤ ′ ′ generality that ℓ2i−1 B1,1 and ℓ2i B1,2 for 1 i k, and that we meets ℓ1, . . . , ℓn in as we go along Fix(G1).

Suppose ℓ2i−1 ⊂ B2,2 and ℓ2i ⊂ B2,1 for some i. By taking a proper arc on B/G2 connecting ℓ2i/G2 and Fix(G2)/G2, we obtain a setwise G2-invariant arc α on B which meets Fix(G2) in a point and con- nects ℓ2i−1 and ℓ2i. Then Fix(G2) is modified by the half twist along the loop ∂N(ℓ2i−1 ∪ ℓ2i ∪ α) ∩ intB, denoted by λ, so that afterwards ℓ2i−1 ⊂ B2,1 and ℓ2i ⊂ B2,2, as illustrated in Figure 2. The argument presented for the proof of Lemma 2.1 implies that this modification is realized by a 1/2-twist along the −1 3 torus p (λ) which conjugates G2 to a subgroup of Diff(S ). We may therefore assume ℓ2i−1 ⊂ B2,1 and ℓ2i ⊂ B2,2 for 1 ≤ i ≤ k.

Figure 2. Half twist along λ.

¸ `2i¡1

Fix(G2) ®

`2i

′ ′ ′ ∩ Suppose that ℓi and ℓj are connected by an arc α in Fix(G2) B. Then Fix(G2) is modified by the ′ ′ ∪ ′ ∪ ′ ∩ ′ ′ half twists along the loop λ = ∂N(ℓi ℓj α ) intB so as to meet ℓi and ℓj in the reverse order, as −1 ′ illustrated in Figure 3, which is realized by the conjugation of G2 by a 1/2-twist along the torus p (λ ), { ′ ′ } as before. Since every permutation on the set ℓ2, . . . , ℓn is a product of transpositions, we may assume ′ ′ that Fix(G2) meets ℓ1, . . . , ℓn in order. Moreover, we can change the order in which Fix(G2) meets the ′ ∩ ′ two points in ℓi Fix(G2) by the half twists along ℓi, which is also realized by a 1/2-twist along the −1 ′ torus p (ℓi). We may therefore assume that G2 is equivalent to G1 relative to ∂B.

Figure 3. Half twist along λ′.

`0 `0 i j

Fix(G2) ¸0 ®0

Now we may assume G1 = G2. Take a map h ∈ Diff(M) which restricts to the identity map on F 1 and takes P2 to P1 setwise preserving every S -fiber. It is easy to verify that h is extendable to a map 3 in Diff(S ). Hence, the conclusion follows by conjugating G2 by h.

Lemma 2.3. Let M be a compact connected 3-manifold in S3 with non-empty boundary whose interior admits a complete hyperbolic structure of finite volume, and F a non-empty closed submanifold of ∂M. 3 Suppose that G1 and G2 are finite group actions on S such that Symmetry 2012, 4 32

(1) G1(M) = G2(M) = M and G1(F ) = G2(F ) = F ,

(2) G1 and G2 induce the same permutation on the set of the components of ∂M, and

(3) G1 and G2 agree on F .

Then there is a sequence of rational twists along tori in F whose composition conjugates G2 to a finite b b group action G2 such that the actions of G1 and G2 on M is equivalent relative to F .

Proof. It follows from Theorem 5.5 of [18] that intM admits two complete hyperbolic structures of

finite volume, one is G1-invariant and the other is G2-invariant. Mostow’s rigidity theorem [15] implies that complete hyperbolic structures of finite volume on intM are unique up to isometry representing the identity map on Out(π1(M)). We may therefore assume that intM is endowed with the G1-invariant ′ ∈ hyperbolic structure, and that G2 is conjugate to an isometric action G2 by h Diff(M) which is isotopic to the identity map. Next, we are going to modify h in a regular neighborhood of F so as to restrict to the identity map on

F . It follows from Propostition D.3.18 of [19] that F consists of tori. Let ht be an isotopy from h to the identity map. Denote by G2 the finite group action on F × I whose restriction on F × {t} is induced from the finite group action on F given by the conjugate of G2 by ht. In particular, the actions of G2 on × { } × { } ′ F 0 and F 1 are respectively given by G2 and G2. Note that G2 preserves the product structure 3 3 F × ∂I, and that we can embed F × I in S so that G2 extends to a finite group action on S . We consider the partition of the set of the components of F into the orbits under the permutation induced by G2. Suppose that the orbits are represented by T1,...,Tn. Lemma 2.1 implies that a rational twist along Ti × {1} conjugates the setwise stabilizer of Ti × I in G2 so that the action on Ti × I is equivalent relative to Ti × ∂I to the action which preserves the product structure. Suppose that the rational twists along the tori in F ×{1} are equivariantly induced from those along T1×{1},...,Tn×{1}.

By conjugating G2 by their composition, it is equivalent relative to F × ∂I to the action which preserves the product structure. This implies that h is modified equivariantly so as to restrict to the identity map on F . ∈ ∈ ◦ −1 Suppose that g1 G1 and g2 G2 agree on F . Then g1 g2 restricts to the identity map on F . Since the of intM is finite (see [15]), Newman’s theorem [20] implies g1 = g2. Hence, G1 and ′ G2 agree on M. This completes the proof. Lemma 2.4. Let M be a compact connected 3-manifold in S3 with non-empty boundary such that the double M of M along a non-empty compact submanifold Φ of ∂M admits a complete hyperbolic structure of finite volume in its interior. Let F be a closed submanifold of ∂M containing Φ. Suppose 3 that G1 and G2 are finite group actions on S such that

(1) G1(M) = G2(M) = M and G1(F ) = G2(F ) = F ,

(2) G1 and G2 induce the same permutation on the set of the components of ∂M, and

(3) G1 and G2 agree on F .

Then there is a finite sequence of rational twists along tori in F whose composition conjugates G2 to a b b finite group action G2 such that the actions of G1 and G2 on M are equivalent relative to F . Symmetry 2012, 4 33

Proof. We see M as the quotient of M by Z2 generated by an orientation-reversing involution. Each Gi induces a finite group action Gi on M which is an extension of Z2 by Gi. As in the proof of Lemma 2.3, we consider intM endowed with a G1-invariant hyperbolic structure. Then some h ∈ Diff(M), which is ′ isotopic to the identity map, conjugates G2 to an isometric action G2. Clearly, Φ meets intM in a totally geodesic surface, and therefore h(Φ) = Φ holds. ∈ ∈ ′ ∈ ∈ Suppose that g1 G1 and g2 G2 respectively induce g1 G1 and g2 G2 which agree on F . Then −1 ◦ g1 g2 restricts to an isometry on each component Φi of Φ, which is a compact surface of negative Euler −1 ◦ characteristic (see Propostition D.3.18 of [19]). Since g1 g2 is trivial in Out(π1(Φi)), g1 and g2 agree on Φi. Therefore, [20] implies g1 = g2. Hence, some h ∈ Diff(M), which setwise preserves Φ and is isotopic to the identity map, conjugates the action of G1 on M to G2. It follows from Proposition D.3.18 of [19] that F − Φ consists of tori. As in the proof of Lemma 2.3, modify h in N(F − Φ) by rational twists along tori in F − Φ so that afterwards h restricts to the identity map on F − Φ and conjugates the action of G1 on M to G2. Moreover, we may assume by Lemma 2.3 of [11] that h restricts to the identity map on Φ. Since h extends to an of S3 which is diffeomorphic outside M, the conclusion follows.

Proposition 2.5. Theorem 1.1 is true, if Γ is non-splittable.

Proof. The equivariant loop theorem (see Chapter VII of [15] and [21]) implies that there is a

G1-invariant system D1 of disjoint disks properly embedded in E(Γ ) which splits E(Γ ) into pieces with incompressible boundary. The equivariant Dehn’s lemma [21,22] implies that the boundary loops of D1 bound a G2-invariant system D2 of disjoint disks properly embedded in E(Γ ). Since Γ is non-splittable, E(Γ ) is irreducible. Therefore, there is an isotopy of E(Γ ) relative to ∂E(Γ ) which 2 takes D2 to D1. Since any finite group action on D is orthogonal [15], we may assume that G1 and

G2 agree on D1. Moreover, the induced actions on the balls obtained by splitting E(Γ ) along D1 are equivalent relative to the boundary (see [15]). Therefore, it is enough to consider the case where E(Γ ) is a Haken manifold with incompressible boundary. We may assume by the equivariant JSJ decomposition theorem (see Theorem 8.6 of [16]) and by the uniqueness of the JSJ decomposition [13,14] that there is a G1-invariant and G2-invariant system T of essential annuli and tori in E(Γ ) realizing the canonical JSJ decomposition of the pair (E(Γ ), ∂E(Γ )). The argument presented for the proof of Proposition 3.10 of [11] implies that some h ∈ Diff(S3), which is isotopic to the identity map relative to N(Γ ), conjugates G2 to a finite group action which agree with G1 on the annuli in T . We may therefore assume that T contains no annuli.

The rest of the proof proceeds by induction on the number of tori in T . Take a piece Mk attaching ∂E(Γ ). By Lemmas 2.2, 2.3 and 2.4, it is enough to consider the case where G2 agrees with G1 on G1(Mk). Moreover, we may assume by Lemma 2.1 that G1 and G2 agree on the components of cl(E(Γ ) − G1(Mk)) each of which is a product I-bundle over a torus. Hence, the conclusion follows by the induction hypothesis.

3. Possibly Splittable Case

For a symmetry group G of a splittable spatial graph Γ in S3, there is a setwise G-invariant system S of spheres realizing the prime factorization of E(Γ ) (see [23]). However, S is not unique in contrast Symmetry 2012, 4 34 to the JSJ decomposition of a Haken 3-manifold. If some component of Γ is setwise invariant and every essential sphere in E(Γ ) has a trivial stabilizer, there is a canonical choice of S (see [11]). We first prove that this is possible also in the setting of Theorem 1.1.

3 Lemma 3.1. Let Γ be a splittable spatial graph in S . Suppose that G1 and G2 are symmetry groups of Γ such that

(1) G1(γ) = G2(γ) = γ for at least one component γ of Γ ,

(2) G1 and G2 are cyclic groups acting on Γ freely, and

(3) G1 and G2 agree on N(Γ ).

3 Then each Gi admits a setwise Gi-invariant system Bi of disjoint balls in S not containing γ such that each ∂Bi realizes the prime factorization of E(Γ ). Moreover, for some choice of B1 and B2, there is a 3 finite sequence of rational twists along incompressible tori in E(Γ )−intB2 and a map in Diff(S ) which 3 restricts to the identity map on N(Γ ) whose composition conjugates the action of G2 on S − intB2 to 3 the action of G1 on S − intB1.

Proof. Denote by Γγ the non-splittable spatial subgraph of Γ containing γ which is obtained by the prime factorization of E(Γ ). It follows from the equivariant sphere theorem [23] that each Gi admits a setwise Gi-invariant system Si = Si,1 ∪ · · · ∪ Si,n of disjoint, non-parallel, essential spheres in E(Γ ) realizing the prime factorization. Suppose that each Si,j bounds a ball Bi,j disjoint from γ. Note that

Sing(Gi) avoids Bi,j or meets Bi,j in a trivial 1-string tangle (see [15]). 3 3 3 Suppose Bi,j ⊂ Bi,k for some distinct j and k. Denote by pi : S → S /Gi = S the projection map onto the quotient space. Take an arc α properly embedded in pi(Bi,k −intBi,j) which connects pi(∂Bi,k) and pi(∂Bi,j). Suppose that α lies on pi(Sing(Gi)) if Sing(Gi) connects ∂Bi,j and ∂Bi,k. By replacing

Bi,j with another ball in intBi,k if necessary, α meets Si in its endpoints. By drilling into pi(Bi,k) along

α ∪ pi(Bi,j), Bi,k is deformed to a ball disjoint from Bi,j, as illustrated in Figure 4 in which the result of the deformation is presented in a cross-sectional view. By a finite repetition of this operation, we obtain a system Bi = Bi,1 ∪ · · · ∪ Bi,n of disjoint balls. This proves the first half of the lemma. Without loss of generality, Γ ∩ B2,j = Γ ∩ B1,j for 1 ≤ j ≤ n.

Figure 4. Modification of pi(Bi,k) which makes pi(Bi,k) disjoint from pi(Bi,j).

pi(Bi;j ) pi(Bi;j ) ®

pi(Bi;k)

Proposition 2.5 implies that there is a finite sequence of rational twists along incompressible tori in b E(Γγ) whose composition h conjugates G2 to a symmetry group G2 of Γγ equivalent to G1 relative to

N(Γγ). By a G2-equivariant isotopy, we may assume that these incompressible tori are disjoint from 3 B2. Then h restricts to the identity map on B2. Suppose that H ∈ Diff(S ,Γγ) realizes the above b b equivalence of G2 and G1. Then H takes Sing(G2) to Sing(G1). As a consequence of the affirmative Symmetry 2012, 4 35

answer to the Smith conjecture [15], Sing(G1) is either an empty set, a trivial knot, or a Hopf link whose components have different indices. Suppose that the orientation of Sing(G1) is induced from the b orientation of Sing(G2) by H. b b Suppose that B2,j and B2,k are connected by an arc β in Sing(G2) − intB2, and that Sing(G2) meets b B2,k in an arc δ. Then B2,k can be modified by a G2-equivariant deformation along β ∪B2,j similar to the inverse of that mentioned above so as to contain B2,j. Moreover, it can be deformed along β ∪ δ ∪ B2,j so as to avoid B2,j again, as illustrated in Figure 5. Note that this operation changes the order in which b the circle in Sing(G2) containing β meets the balls in {B2,1,...,B2,n}.

Figure 5. Modification of B2,k realizing the transposition of B2,j and B2,k.

±

B2 ¯ ;j ¯ [ ± B2;j B2;k

Let C be a component of Sing(G1). Without loss of generality, C meets B1,1,...,B1,r in order, and −1 b avoids B1,r+1,...,B1,n. Since G1 and G2 agree on N(Γ ), the component H (C) of Sing(G2) meets

B2,1,...,B2,r possibly not in order. Since every permutation on the set {B2,1,...,B2,r} is a product −1 of transpositions realized by the above operation, we may assume that H (C) meets B2,1,...,B2,r in order. Apply this argument to each component of Sing(G1). Since each ∂Bi realizes the prime factorization of E(Γ ), we can modify H by a G1-equivariant isotopy relative to N(Γγ) so that we have b H(Sing(G2)) = Sing(G1) and H(B2,j) = B1,j for each j. Thus, H is modified so as to conjugate the b 3 3 action of G2 on S − intB2 to the action of G1 on S − intB1.

After this modification, H restricts to an orientation-preserving homeomorphism on B1. Therefore,

H(Γ ∩ B2) is ambient isotopic to Γ ∩ B1 in B1. Hence, H can be modified in B1 so as to restrict to the identity map on N(Γ ). This completes the proof.

2 Lemma 3.2. Suppose that G1 and G2 are orientation-preserving finite cyclic group actions on S × I such that

2 (1) G1 and G2 do not interchange the components of S × ∂I, and

2 (2) G1 and G2 agree on S × ∂I.

2 Then a rational twist along S × {1} conjugates G2 to a finite group action equivalent to G1 relative to S2 × ∂I.

Proof. It is enough to consider the case where G1 is not trivial. It follows from the remark after Theorem 2 8.1 of [16] that S × I admits a G1-invariant product structure P1 and a G2-invariant product structure 2 2 P2. Since the actions of G1 and G2 on S ×{0} are conjugate to a rotation of S (see [15]), each Fix(Gi) 2 consists of two I-fibers in Pi. Since G1 and G2 agree on S × ∂I, we have ∂Fix(G1) = ∂Fix(G2). 2 2 Denote by pi : S × I → S × I/Gi the projection map onto the quotient space for each i, and by St 2 2 the S -fiber S × {t} in P1. Connect the two cone points of p1(S0) by an arc a embedded in p1(S0). Symmetry 2012, 4 36

−1 Then p1 (¯a) is a spatial θn-curve consisting of two vertices on the fixed points and n > 1 edges each P −1 connecting them. Denote by Ai the branched surface consisting of I-fibers in i attaching p1 (¯a) for each i. Then each p1(Ai ∩ S1) is an arc connecting the two cone points on p1(S1). Since the underlying space of p1(S1) is a sphere, p1(A2 ∩ S1) is isotopic to p1(A1 ∩ S1) relative to the cone points. Therefore,

A2 is deformed by a G2-equivariant isotopy relative to S0 so that A1 ∩ S1 = A2 ∩ S1. There are two cases depending on whether Fix(G1) and Fix(G2) are isotopic relative to the endpoints or not.

Assume that Fix(G1) and Fix(G2) are isotopic relative to the endpoints. Then P2 is deformed by 2 an isotopy relative to S × ∂I so as to agree with P1 on a setwise G1-invariant tubular neighborhood

N(Fix(G1)) saturated in the I-bundle structure induced from P1. Since each Ai meets the solid torus 2 (S × I) − intN(Fix(G1)) in the system of meridian disks, A2 is moved to A1 by an isotopy relative to 2 (S ×∂I)∪N(Fix(G1)). We may therefore assume A1 = A2, and that G1 and G2 agree on N(Fix(G1)).

Then the I-bundle structures in P1 and P2 respectively induce the orbifold isomorphisms φ1 : p2(S1) → p2(S0) and φ2 : p2(S0) → p1(S1) such that φ2 ◦ φ1 is isotopic to the identity map by an isotopy relative to the cone points which setwise preserves p2(A2 ∩ S1). Then we can deform P2 by an isotopy on 2 p2(S × I) relative to p2(S0) which setwise preserves p2(A2 ∩ S1) so that P1 and P2 induce the same 1 2 ∂I-bundle structure on S × ∂I. Hence, the diffeomorphism of S × I which takes P2 to P1 induces 2 the equivalence of G1 and G2 relative to S × ∂I, as required.

Assume that Fix(G1) and Fix(G2) are not isotopic relative to the endpoints. Let h: S1 → S1 be a lift of an orientation-preserving involution on p1(S1) which interchanges the cone points. Then h is a diffeomorphism isotopic to the identity map which conjugates the action of G2 on S1 to itself and is realized by a 1/2-twist along the sphere S1. We may therefore assume that Fix(G1) and Fix(G2) are isotopic relative to the endpoints. Hence, the conclusion follows by the argument presented for the previous case. Proof of Theorem 1.1. It is enough by Proposition 2.5 to prove the theorem in the case where Γ is splittable. Then G1 and G2 are cyclic groups acting on Γ freely. We may assume by Lemma 3.1 that 3 there is a setwise G1-invariant and setwise G2-invariant system B of disjoint balls in S not containing

γ such that ∂B realizes the prime factorization of E(Γ ), and that G1 and G2 agree on E(B).

Suppose that B consists of balls B1,...,Bn. Each Γ ∩ Bi is a non-empty, non-splittable, spatial subgraph of Γ . By applying Proposition 2.5 to the actions of the setwise stabilisers of Bi in G1 and G2 on Bi, we may assume that G1 and G2 agree on E(∂B). Hence the conclusion follows by applying

Lemma 3.2 to the actions of G1 and G2 on N(∂B) equivariantly.

Remark 3.3. Theorem 1.1 requires the spatial graph Γ to have no companion knot, and the symmetry groups G1 and G2 of Γ to act on Γ freely if Γ is splittable. These requirements are needed because of the following examples.

(1) Suppose that Γ is a granny knot. Then Γ has two companion knots K1 and K2, both of which are

trefoil knots. We obtain E(K1), E(K2), and a 2-fold composing space by the JSJ decomposition

of E(Γ ). Figure 6 illustrates Z2-symmetries G1 and G2 of Γ such that G2 interchanges E(K1) 3 and E(K2) but G1 does not. By conjugating G1 by a map in Diff(S ) which moves N(Γ ) in the

longitudinal direction, G1 and G2 are not equivalent but agree on ∂N(Γ ). Moreover, any rational Symmetry 2012, 4 37

twists along incompressible tori in E(Γ ) cannot change the induced symmetries of E(K1) and

E(K2), since the trefoil knot exterior is atoroidal.

Figure 6. Z2-symmetries of a spatial graph with companion knots.

G2

G1

(2) Suppose that Γ is a spatial graph which splits into non-splittable spatial graphs γ1, γ2 and γ3, as

illustrated in Figure 7, where γ1 is a spatial θ-curve. According to the choice of two edges of γ1, we

obtain a trefoil knot K1, a figure-eight knot K2, or their connected sum K1#K2. Then any map in 3 Diff(S ,Γ ) does not permute these edges. The Z2-symmetries G1 and G2 of Γ illustrated in Figure 3 7 are not equivalent, since there is no map in Diff(S ,Γ ) which takes Sing(G1) to Sing(G2) and

interchanges γ2 and γ3. Moreover, we cannot perform rational twists along incompressible spheres

and tori in E(Γ ) to make G2 equivalent to G1, since any setwise G2-invariant incompressible

sphere in E(Γ ) separates γ2 and γ3.

Figure 7. Z2-symmetries which are not free on a splittable spatial graph.

G1 G2

Acknowledgements

The author would like to thank the referees for helpful comments which improved this paper.

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